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A276424 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its even singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once.

%I A276424 #22 Mar 07 2025 06:29:32
%S A276424 1,1,0,1,0,1,2,0,1,0,3,0,1,0,1,4,0,2,0,1,0,6,0,2,0,1,0,2,8,0,3,0,2,0,
%T A276424 2,0,11,0,4,0,3,0,2,0,2,15,0,5,0,4,0,4,0,2,0,19,0,7,0,6,0,5,0,2,0,3,
%U A276424 25,0,9,0,8,0,7,0,4,0,3,0,34,0,11,0,10,0,10,0,5,0,3,0,4
%N A276424 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its even singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once.
%C A276424 T(n,0) = A265254(n).
%C A276424 T(n,n) = A035457(n).
%C A276424 Sum_{k>=0} k*T(n,k) = A276425(n).
%C A276424 Sum_{k>=0} T(n,k) = A000041(n).
%H A276424 Alois P. Heinz, <a href="/A276424/b276424.txt">Rows n = 0..200, flattened</a>
%F A276424 G.f.: G(t,x) = Product_{j>=1} ((1-x^(2*j))*(1+t^(2*j)*x^(2*j)) + x^(4*j))/(1-x^j).
%e A276424 Row 4 is 3, 0, 1, 0, 1 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the even singletons are 0, 2, 0, 0, 4, respectively.
%e A276424 Row 5 is 4, 0, 2, 0, 1, 0 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the even singletons are 0, 2, 0, 0, 2, 4, 0, respectively.
%e A276424 Triangle starts:
%e A276424   1;
%e A276424   1,0;
%e A276424   1,0,1;
%e A276424   2,0,1,0;
%e A276424   3,0,1,0,1;
%e A276424   4,0,2,0,1,0;
%e A276424   6,0,2,0,1,0,2.
%p A276424 g := Product(((1-x^(2*j))*(1+t^(2*j)*x^(2*j))+x^(4*j))/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 23)): for n from 0 to 20 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, i), i = 0 .. n) end do; # yields sequence in triangular form
%p A276424 # second Maple program:
%p A276424 b:= proc(n, i) option remember; expand(
%p A276424       `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
%p A276424       `if`(j=1 and i::even, x^i, 1), j=0..n/i))))
%p A276424     end:
%p A276424 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A276424 seq(T(n), n=0..14);  # _Alois P. Heinz_, Sep 14 2016
%t A276424 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*If[j == 1 && EvenQ[i], x^i, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 11 2016, after _Alois P. Heinz_ *)
%Y A276424 Cf. A000041, A035457, A265254, A276422, A276423, A276425.
%K A276424 nonn,tabl
%O A276424 0,7
%A A276424 _Emeric Deutsch_, Sep 14 2016